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5.1a Areas and Distances

5.1a Areas and Distances. State Standard – 16.0a Students use definite integrals in problems involving area. Objective – To be able to use the 2 nd derivative test to find concavity and points of inflection. Area. We can easily calculate the exact area of regions with straight sides.

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5.1a Areas and Distances

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  1. 5.1a Areas and Distances State Standard – 16.0a Students use definite integrals in problems involving area. Objective – To be able to use the 2nd derivative test to find concavity and points of inflection.

  2. Area We can easily calculate the exact area of regions with straight sides. For Instance: w h l b A = lw A = ½bh However, it isn’t so easy to find the area of a region with curved sides.

  3. Circumscribed rectangles are all above the curve: Upper sum Inscribed rectangles are all below the curve: Lower Sum

  4. Lower Sum Upper Sum

  5. Let’s find the area under the curve by using rectangles using right endpoints. y = x2 from [0,4] n = 4 10 9 8 7 6 What is the Area of each rectangle? 5 4 3 A = (1)(12) +(1)(22) +(1)(32) +(1)(42) 2 1 A =1 + 4 + 9 + 16 –3 –2 –1 1 2 3 4 A =30 sq units

  6. Let’s find the area under the curve by using rectangles using left endpoints. y = x2 from [0,4] n = 4 10 9 8 7 What is the Area of each rectangle? 6 5 4 A = (1)(02) +(1)(12) +(1)(22) +(1)(32) 3 2 A =0 + 1 + 4 + 9 1 A =14 sq units –3 –2 –1 1 2 3 4

  7. 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 –3 –2 –1 1 2 3 4 –3 –2 –1 1 2 3 4 A =(30 + 14) / 2 A =30 sq units A =14 sq units A =22 sq units

  8. Let’s find the area under the curve by using rectangles using left endpoints. y = x2 from [0,4] n = 8 10 9 8 7 What is the Area of each rectangle? 6 5 4 A = + + + 3 2 + + + + 1 –3 –2 –1 1 2 3 4 =17.5 sq units

  9. 5.1a Homework Approximate the area from (a) the left side (b) the right side 1) 2) 3)

  10. 5.2a The Definite Integral State Standard – 13.0a Students know the definition of the definite integral using by using Riemann sums. Objective – To be able to use definite integrals to find the area under a curve.

  11. Let’s find the area under the curve by using rectangles using right endpoints. We could also use a Right-hand Rectangular Approximation Method (RRAM). y = x2 from [0,4] 10 n = 4 9 8 7 6 What is the Area of each rectangle? 5 4 3 A =(1)(42) +(1)(32) +(1)(22) +(1)(12) 2 1 A =16 + 9 + 4 + 1 –3 –2 –1 1 2 3 4 A =30 sq units

  12. Let’s find the area under the curve by using rectangles using left endpoints. This is called the Left-hand Rectangular Approximation Method (LRAM). y = x2 from [0,4] n = 4 10 9 8 7 What is the Area of each rectangle? 6 5 4 A = (1)(02) +(1)(12) +(1)(22) +(1)(32) 3 2 A =0 + 1 + 4 + 9 1 A =14 sq units –3 –2 –1 1 2 3 4

  13. Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). y = x2 from [0,4] n = 4 10 9 8 7 6 What is the Area of each rectangle? 5 4 3 A =(1)(0.52) +(1)(2.52) +(1)(1.52) +(1)(3.52) 2 1 A =0.25 + 2.25 + 6.25 + 12.25 –3 –2 –1 1 2 3 4 A =21 sq units

  14. Mid-Point Rule: ( h = width of subinterval )

  15. Approximate area: Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

  16. Approximate area: With 8 subintervals: width of subinterval

  17. 5.2a Homework Approximate the area from the midpoint MRAM 1) 2) 3)

  18. 5.2b The Definite Integral State Standard – 13.0a Students know the definition of the definite integral using by using Riemann sums. Objective – To be able to use definite integrals to find the area under a curve.

  19. 10 10 9 9 8 8 10 7 7 9 6 6 8 5 5 7 4 4 6 3 3 5 2 2 4 1 1 3 –3 –2 –1 1 2 3 4 –3 –2 –1 1 2 3 4 2 1 –3 –2 –1 1 2 3 4 LRAM RRAM Right Rectangular Approximation Method MRAM

  20. It is called a dummy variable because the answer does not depend on the variable chosen. Definite Integrals DefinitionArea Under a Curve (as a Definite Integral) If y = f(x) is nonnegative and can be integrated over a closed interval [a,b], then the area under the curve y = f(x) from a to b is the integral of f from a to b, upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration

  21. If the velocity varies: Distance: After 4 seconds: The distance is still equal to the area under the curve! Notice that the area is a trapezoid.

  22. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 2. Constant multiples can be moved outside. 3. Properties of Definite Integrals Integrals can be added and subtracted.

  23. 5. Integrals can be separated

  24. Given

  25. 5.2b Homework p. 391 # 21 – 25 all

  26. 5.3 The Fundamental Theorem of Calculus Morro Rock, California

  27. The First Fundamental Theorem of Calculus If f is continuous at every point of , and if F is any antiderivative of f on , then (Also called the Integral Evaluation Theorem)

  28. When finding indefinite integrals, we always include the “plus C”. Integrals such as are called indefinite integrals because we can not find a definite value for the answer.

  29. Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

  30. Find the area under the curve from x=1 to x=2. Area under the curve from x=1 to x=2. Area from x=0 to x=2 Area from x=0 to x=1

  31. Actual area under curve:

  32. Find the area between the x-axis and the curve from to . pos. neg.

  33. 5.3 Homework Pg 402 19 – 30 and 35

  34. 5.3b The 2nd Fundamental Theorem of Calculus Morro Rock, California

  35. The 2nd Fundamental Theorem of Calculus If f is continuous on , then the function has a derivative at every point in , and

  36. 1. Derivative of an integral. Second Fundamental Theorem:

  37. Second Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration.

  38. Second Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

  39. Second Fundamental Theorem: New variable. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

  40. The long way: Second Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

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